MATHEMATICS

Yu. P. GOR'KOV

ON THE BEHAVIOR OF SOLUTIONS OF BOUNDARY-VALUE PROBLEMS FOR A QUASILINEAR PARABOLIC EQUATION AS \(t \to \infty\)

(Presented by Academician I. G. Petrovskii, March 9, 1964)

1. In the present note we study the behavior, as \(t \to \infty\), of the solution of the parabolic equation

\[ Lu \equiv \frac{D}{Dx} f(x,u,u_x) - a(x,t,u,u_x)\frac{\partial u}{\partial t} = 0 \tag{1} \]

(the symbol \(D/Dx\) denotes total differentiation with respect to \(x\)) for the boundary-value problem:

\[ u(x,0)=u_0(x), \qquad u(0,t)=0, \qquad u_x(l,t)=A_0, \tag{2} \]

and also for the first boundary-value problem:

\[ u(x,0)=u_0(x), \qquad u(0,t)=\varphi_1(t), \qquad u(l,t)=\varphi_2(t) \tag{3} \]

under the condition that \(\varphi_i(t)\to \varphi_i^0\) as \(t\to\infty\) \((i=1,2)\).

Suppose that, for all \(u\) and \(p\) and \((x,t)\in D\{0\le x\le l,\;0\le t<\infty\}\), the following conditions are satisfied:

\[ \frac{\partial f(x,u,p)}{\partial p} \ge \alpha > 0, \tag{4} \]

\[ 0<a(x,t,u,p)\le \gamma_1(x,u,p), \tag{5} \]

where \(\alpha\) is a constant and \(\gamma_1(x,u,p)\) is a certain continuous function.

It is natural to expect that the solution \(u(x,t)\), as \(t\to\infty\), tends to the solution \(z(x)\) of the ordinary differential equation

\[ \frac{D}{Dx} f(x,z,z_x)=0, \tag{6} \]

or, what is the same thing, to the solution \(z(x,c)\) of the equation

\[ f(x,z,z_x)=c \tag{6'} \]

for some fixed value of the parameter \(c\).

1. In considering the boundary-value problem (1), (2), we shall assume that the following conditions are fulfilled:

a) the function \(f(x,u,p)\) is six times continuously differentiable with respect to \(x,u,p\); the function \(a(x,t,u,p)\) is five times continuously differentiable with respect to \(x,t,u,p\) (for \((x,t)\in D\) and arbitrary \(u\) and \(p\));

b) there exists a solution \(u(x,t)\) of problem (1), (2), and the derivative \(u_x(x,t)\) is bounded in the domain \(D_T\{0<x<l,\;0<t<T\}\) for every \(T>0\).

Suppose now that, for the given \(A_0\), there exists a solution \(v(x)\) of equation (6) satisfying the conditions: \(v(0)=0,\; v'(l)=A_0\). In this case, on the interval \([0,l]\), for all \(c\) from some neighborhood \((c_1,c_2)\) of the point \(c_0\equiv f(x,v(x),v'(x))\), there exists a solution \(z(x,c)\) of the Cauchy problem for the equa-

…of equation \((6')\) with the condition \(z(0,c)=0\). Suppose, moreover, that

\[ z_{xc}(l,c_0)>0 \tag{7} \]

and that the initial function \(u_0(x)\) satisfies the inequalities

\[ z(x,\tilde c_1)\leq u_0(x)\leq z(x,\tilde c_2) \tag{8} \]

(here \(\tilde c_1,\tilde c_2\in(c_1,c_2)\) and are such that \(z_{xc}(l,c)>0\) for \(\tilde c_1<c<\tilde c_2\)).

Theorem 1. If conditions (4), (5), a), b), (7), and (8) are satisfied, then the solution of problem (1), (2) satisfies \(u(x,t)\to v(x)\) as \(t\to\infty\), uniformly in \(x\).

Proof. Denote by \(w_i(x,t)\) the solutions of equation (1) satisfying the conditions: \(w_i(x,0)=z(x,\tilde c_i)\), \(w_i(0,t)=0\),

\[ \frac{\partial}{\partial x}w_i(l,t)=A_i(t)\quad (i=1,2). \]

The functions \(A_i(t)\in C^6[0,\infty)\) and are subject to the conditions:

\[ A_1'(t)\geq0,\qquad A_2'(t)\leq0,\qquad \lim_{t\to\infty}A_i(t)=A_0,\qquad A_i(t)\equiv z_x(l,\tilde c_i) \]

for \(0\leq t\leq\delta\) \((\delta>0)\).

We shall show that the following inequalities hold in the domain \(D\):

\[ w_1(x,t)\leq u(x,t)\leq w_2(x,t),\qquad z(x,\tilde c_1)\leq w_i(x,t)\leq z(x,\tilde c_2), \tag{9} \]

\[ \frac{\partial}{\partial t}w_1(x,t)\geq0,\qquad \frac{\partial}{\partial t}w_2(x,t)\leq0\quad (i=1,2). \]

Represent equation (1) in the form

\[ a(x,t,u,u_x)\frac{\partial u}{\partial t} = b(x,u,u_x)\frac{\partial^2u}{\partial x^2} + d(x,u,u_x)\frac{\partial u}{\partial x} + c(x,u)u+g(x), \tag{1'} \]

where

\[ b(x,u,u_x)=f_p(x,u,u_x),\quad d(x,u,u_x)=\int_0^1 f_{xp}(x,u,\tau u_x)\,d\tau+f_u(x,u,u_x), \]

\[ c(x,u)=\int_0^1 f_{xu}(x,\tau u,0)\,d\tau,\quad g(x)=f_x(x,0,0). \]

Since the functions \(w_i(x,t)\), \(u(x,t)\), and \(z(x,\tilde c_i)\) \((i=1,2)\) are solutions of equation \((1')\), the difference of any two of them, as well as the functions \(\frac{\partial}{\partial t}w_i(x,t)\), may be regarded as solutions of certain linear homogeneous parabolic equations with coefficients continuous inside \(D\).

In order to obtain inequalities (9), it is enough to use Theorems 6 and 7 of [1].

We next show that in the domain \(D\) the estimate

\[ \left|\frac{\partial w_i}{\partial x}\right|\leq M, \tag{10} \]

holds, where \(M\) is some constant.

Integrating the identity \(Lw_i=0\) with respect to \(x\) from \(x\) to \(l\), we obtain the relation:

\[ \int_x^l a\left(x,t,w_i,\frac{\partial}{\partial x}w_i\right)\frac{\partial w_i}{\partial t}\,dx = f(l,w_i(l,t),A_i(t))- f\left(x,w_i,\frac{\partial}{\partial x}w_i\right). \tag{11} \]

In view of the boundedness of

\[ \left|f\left(0,0,\frac{\partial}{\partial x}w_i(0,t)\right)\right| \]

and the inequality

\[ \left|\int_x^l a\,\frac{\partial w_i}{\partial t}\,dx\right| \leq \left|\int_0^l a\,\frac{\partial w_i}{\partial t}\,dx\right|, \]

from (11) it follows that there is a bound, uniform in \(x\) and \(t\) in the domain \(D\), …

of the functions \(f\left(x,w_i,\dfrac{\partial}{\partial x}w_i\right)\) \((i=1,2)\). Estimate (10) now follows from inequalities (4) and (9). Denote
\[ \lim_{t\to\infty} w_i(x,t)=\mu_i(x). \]
Since the integral
\[ \int_0^t\int_0^l a\left(x,t,w_i,\frac{\partial}{\partial x}w_i\right)\frac{\partial w_i}{\partial t}\,dx\,dt \]
is bounded uniformly in \(t\) for \(t\geqslant 0\), for each \(i\) \((i=1,2)\) there exists a sequence of numbers \(\{t_n^i\}\) such that
\[ \int_0^l a\,\frac{\partial w_i}{\partial t}\,dx\to 0 \quad\text{as } t_n^i\to\infty. \]
Hence it follows that
\[ \int_x^l a\,\frac{\partial w_i}{\partial t}\,dx\to 0 \quad\text{as } t_n^i\to\infty \]
uniformly with respect to \(x\). Taking this into account, from relation (11) we obtain:
\[ f\left(x,w_i(x,t_n^i),\frac{\partial}{\partial x}w_i(x,t_n^i)\right) \to f\left(x,\mu_i(l),A_0\right) \tag{12} \]
as \(t_n^i\to\infty\), uniformly with respect to \(x\).

The Arzelà theorem may be applied to the sequence of functions \(\{w_i(x,t_n^i)\}\); by virtue of it, for each \(i\) \((i=1,2)\) there exists a subsequence of numbers \(\{t_{n_k}^i\}\) such that
\[ w_i(x,t_{n_k}^i)\to \mu_i(x) \]
as \(t_{n_k}^i\to\infty\), uniformly with respect to \(x\) for \(0\leqslant x\leqslant l\). It then follows from conditions (4) and (12) that the sequence of functions
\[ \left\{\frac{\partial}{\partial x}w_i(x,t_{n_k}^i)\right\} \]
converges uniformly in \(x\) as \(t_{n_k}^i\to\infty\).

Thus \(\mu_i(x)\) are solutions of equation (6), satisfying the conditions:
\[ \mu_i(0)=0,\qquad \mu_i'(l)=A_0. \]
Since \(z(x,c)\) is a monotonically increasing function of \(c\) for a fixed value \(x\in(0,l]\), it follows that
\[ \mu_i(x)\equiv v(x)\quad (i=1,2). \]
The theorem is proved.

Remark 1. In the proof of Theorem 1, the existence of solutions \(w_i(x,t)\) and their smoothness were assumed (continuity of
\[ \frac{\partial}{\partial t}w_i(x,t) \]
in the domain \(D\), and the existence of
\[ \frac{\partial^2}{\partial t^2}w_i(x,t),\qquad \frac{\partial^3}{\partial t\,\partial x^2}w_i(x,t) \]
inside \(D\), \(i=1,2\)). Applying the method of proof of Theorem 1 from paper (²), one can show that the solutions \(w_i(x,t)\) exist and possess the smoothness indicated above.

Remark 2. In the particular case when equation (1) takes the form
\[ \frac{\partial^2u}{\partial x^2} = A(x,t,u)\frac{\partial u}{\partial t} + B(x,u)\frac{\partial u}{\partial x} + F(x,u), \]
assumption b) will be valid if, for all values of \(u\) and \((x,t)\in D\), the following conditions are satisfied: the functions \(A(x,t,u)\), \(B(x,u)\), \(F(x,u)\) have continuous derivatives of third order with respect to all arguments;
\[ 0<A_1(x,t)\leqslant A(x,t,u)\leqslant A_2(x,t), \]
\[ F_u(x,u)\geqslant M_1,\qquad |B_u(x,u)|+|F_{uu}(x,u)|\leqslant M_2; \]
\[ u_0(x)\in C^3[0,l],\qquad u_0(0)=0,\qquad u_0'(l)=A_0 \]
(here \(A_i(x,t)\) are certain continuous functions, \(M_i\) are constants, \(i=1,2\)).

Theorem 1, in essence, asserts that condition (7) is sufficient for the stability of the stationary solution of equation (1). It is proved analogously that from the condition \(z_{xc}(l,c_0)<0\) there follows instability of the stationary solution. If, however, \(z_{xc}(l,c_0)=0\), then the stationary solution may be either stable or unstable.

We now consider the case when, for a given \(A_0\), there exist two solutions of equation (6) satisfying the conditions:
\[ z(0)=0,\qquad z'(l)=A_0. \]
Denote these solutions by \(z_1(x)\) and \(z_2(x)\). Suppose that for all \(c\in[c_1,c_2]\) (where \(c_i=f(x,z_i,z_i')\), \(i=1,2\)) on the interval \([0,l]\) there exists a solution \(z(x,c)\) of the Cauchy problem for equation \((6')\) with the condition \(z(0,c)=0\), and moreover let
\[ z_x(l,c)<A_0 \tag{13} \]

for \(c_1<c<c_2\). Let \(u(x,t)\) denote the solution of problem (1), (2) under the condition that \(u_0(x)\) satisfies the inequalities:

\[ z(x,c_1)<u_0(x)\leq z(x,c_2),\qquad z_x(0,c_1)<u_0'(0)\quad (0<x\leq l). \tag{14} \]

Theorem 2. If conditions (4), (5), a), b), (13), and (14) are fulfilled, then
\[ u(x,t)\to z_2(x)\quad \text{as } t\to\infty \]
uniformly in \(x\).

1. In studying the behavior of the solution of equation (1) in the case of the first boundary-value problem, we shall assume that the following conditions are fulfilled:

c) the existence of solutions \(v_1(x)\) and \(v_2(x)\) of equation (6) such that

\[ \min_{0\leq x\leq l} v_1(x)\geq \max\left\{\max_{0\leq x\leq l}u_0(x),\ \sup_{0\leq t<\infty}\varphi_i(t)\right\}, \]

\[ \max_{0\leq x\leq l} v_2(x)\leq \min\left\{\min_{0\leq x\leq l}u_0(x),\ \inf_{0\leq t<\infty}\varphi_i(t)\right\}\qquad (i=1,2); \]

uniqueness of the solution \(v_0(x)\) satisfying the conditions:
\[ v_0(0)=\varphi_1^0,\qquad v_0(l)=\varphi_2^0; \]

d) the existence of a solution \(u(x,t)\) of equation (1) with derivative \(u_{xx}(x,t)\) bounded in the domain \(D_T\) (for every \(T>0\)) for any sufficiently smooth initial and boundary conditions (3), and its smoothness: continuity of \(u_t(x,t)\) in \(D\) and existence of \(u_{tt}, u_{xxt}\) inside \(D\) in the case when \(u_0(x)\) is a solution of equation (6).

In addition, we shall assume that instead of condition (5) the following condition is fulfilled:

\[ 0<a(x,t,u,p)\leq \gamma_2(x,u) \tag{5'} \]

(where \(\gamma_2(x,u)\) is some continuous function).

Theorem 3. If conditions (4), (5′), c), and d) are fulfilled, then the solution of problem (1), (3)
\[ u(x,t)\to v_0(x)\quad \text{as } t\to\infty \]
uniformly in \(x\).

Remark 3. Sufficient for the fulfillment of condition c) is the following condition:
\[ |f(x,u,0)|\leq M_1+M_2|u|,\qquad M_i>0,\quad 0\leq x\leq l,\quad |u|<\infty\ (i=1,2). \]

Applying Theorem 1 of work \((^3)\), one can indicate additional (very cumbersome) conditions on the functions \(a(x,t,u,p)\), \(f(x,u,p)\), and \(u_0(x)\) under which assertion d) will be valid. The behavior of solutions of boundary-value problems for quasilinear parabolic equations of another type in the case of an arbitrary number of variables was investigated in work \((^4)\).

In conclusion, I take the opportunity to express my deep gratitude to my scientific adviser A. M. Il’in for posing the problem and for his attentive guidance of the work.

Ural State University
named after A. M. Gorky

Received
4 III 1964

REFERENCES

1. A. M. Il’in, A. S. Kalashnikov, O. A. Oleinik, UMN, 17, issue 3 (105), 10—13 (1962).
2. Chjou Yu-lin, Sci. Sinica, 10, No. 8, (1961).
3. A. F. Filippov, DAN, 141, No. 4 (1961).
4. S. N. Khudyaev, Some investigations on nonlinear parabolic and elliptic equations, Dissertation, Moscow State University, 1963.